Definition
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[20F] Let \(Iβ β\), \(x_ 0β \overline{β}\) accumulation point of \(I\), \(f:Iβ β\) function. We define
\begin{align} \limsup _{xβ x_ 0} f(x) = \inf _{U \text{neighbourhood of} x_ 0}~ \sup _{xβ Uβ© I} f(x) \label{eq:limsup_ R}\\ \liminf _{xβ x_ 0} f(x) = \sup _{U \text{neighbourhood of} x_ 0}~ \inf _{xβ Uβ© I} f(x) \label{eq:liminf_ R} \end{align}
where the first βinfβ (resp. the βsupβ) is performed with respect to the family of all the deleted neighbourhoods \(U\) of \(x_ 0\); and the neighbourhoods will be right or left neighbourhoods if the limit is from right or left.